Optimal. Leaf size=350 \[ \frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{b x^2+c x^4}}-\frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{77 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^4 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{77 c^2 \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac{77 c \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}-\frac{11 \sqrt{b x^2+c x^4}}{9 b^2 x^{11/2}}+\frac{1}{b x^{7/2} \sqrt{b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.860347, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{b x^2+c x^4}}-\frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{77 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^4 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{77 c^2 \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac{77 c \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}-\frac{11 \sqrt{b x^2+c x^4}}{9 b^2 x^{11/2}}+\frac{1}{b x^{7/2} \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 83.225, size = 332, normalized size = 0.95 \[ \frac{1}{b x^{\frac{7}{2}} \sqrt{b x^{2} + c x^{4}}} - \frac{11 \sqrt{b x^{2} + c x^{4}}}{9 b^{2} x^{\frac{11}{2}}} + \frac{77 c \sqrt{b x^{2} + c x^{4}}}{45 b^{3} x^{\frac{7}{2}}} + \frac{77 c^{\frac{5}{2}} \sqrt{b x^{2} + c x^{4}}}{15 b^{4} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{77 c^{2} \sqrt{b x^{2} + c x^{4}}}{15 b^{4} x^{\frac{3}{2}}} - \frac{77 c^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{15}{4}} x \left (b + c x^{2}\right )} + \frac{77 c^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{30 b^{\frac{15}{4}} x \left (b + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(5/2)/(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [C] time = 0.23819, size = 210, normalized size = 0.6 \[ \frac{-\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (10 b^3-22 b^2 c x^2+154 b c^2 x^4+231 c^3 x^6\right )-231 \sqrt{b} c^{5/2} x^5 \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+231 \sqrt{b} c^{5/2} x^5 \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )}{45 b^4 x^{7/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)),x]
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Maple [A] time = 0.03, size = 237, normalized size = 0.7 \[{\frac{c{x}^{2}+b}{90\,{b}^{4}} \left ( 462\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{4}b{c}^{2}-231\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{4}b{c}^{2}-462\,{c}^{3}{x}^{6}-308\,b{c}^{2}{x}^{4}+44\,{b}^{2}c{x}^{2}-20\,{b}^{3} \right ){x}^{-{\frac{3}{2}}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(5/2)/(c*x^4+b*x^2)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{6} + b x^{4}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{5}{2}} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(5/2)/(c*x**4+b*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)),x, algorithm="giac")
[Out]